Your task is to draw lines between edges on a regular pentagon such that if you tile a dodecahedron with 12 identical copies of that pentagon you get a single closed line which does not intersect itself and which visits each face of the dodecahedron exactly twice. How many essentially different solutions exist?
2nd Update: I have found a small number of solutions.
I found 6 distinct solutions with a continuous path.
For a line to visit each face twice, there must be 2 entry and 2 exit points.
There are 10 distinct ways to arrange them around a pentagon.
These exclude reflections and rotations.
Moreover, there are two ways for the points to connect without crossing.
Here are the ten face arangements, with the joining variation in vertical pairs.
Only 4 of those 10 sets will make a dodecahedron with matching edges.
And only one (top right) will make a continuous path.
(It's clear on inspection that those with a closed edge loop will not.)
There were 210 possible arrangements.
Only 6 after removing rotational and reflection symmetry.
There are 3 versions of each of the path methods.